Yeah, I applied to a few lower tiered schools but they are still good quality schools that have their pick of outstanding, or at least very solid, applicants. I applied based on fit and tried to shoot for a range of schools to give me at least some options if the top schools don't take me. Honestly though, its all a shot in the dark. I think what it boils down to, assuming you look qualified enough on paper to make any initial cuts, is whether Potential Advisor X takes a shine to your file. So to answer your question, yes, I applied to a few middle-range schools. To show where you applied on your posts, put it in your signature on your account profile.